524 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. 



0! 



Of these the first two, being of the form e~ d f(xat), represent 

 damped waves travelling in the direction of increasing x with the 



velocity a = -^-A . The periodic factor at any time repeats 



its values when x is increased by the distance 1 = \/ ^ 



A V \JJLCO ' 



which is called the wave-length, the frequency being n = o)/2,7r. 



The damping factor e~ An * t "*- x which causes the amplitude of 

 the wave to decrease in geometrical progression as the distance 

 travelled increases in arithmetical, has the relaxation-distance, or 

 the distance in which the amplitude diminishes in the ratio I/e, 



. 



A v27T\fJL(0 



The last two particular solutions represent waves travelling in the 

 opposite direction with the same velocity and damping. 



Since the velocity depends on the frequency, there is no 

 definite velocity of propagation in a conductor. On account of the 

 damping, harmonic disturbances of high frequency rapidly die out, 

 consequently alternating fields penetrate but a short distance into 

 conductors. This was shown by Maxwell*, but its importance 

 was brought out by the researches of Heavisidef, Lord RayleighJ 

 and Hertz . 



We shall now consider the relations between the two fields. 

 If the components X, Y, Z, L, M, N are equal respectively to the 

 complex constants A lt A 2 , A 3 , B l} B 2 , B 3) each multiplied by 

 e it+k* t inserting in the equations (A) and (B) we obtain 



4f7rA\A l = 0, A^ia)B l = 0, 



(12) ^TrAxA^ kB 3 , AfjLia)B 2 = kA 3) 



= kB 2) ApiwB 3 kA 2 . 



Eliminating A 2 /B 3 or A 3 /B 2 we obtain the value for k already 

 found. We thus see that the directional relations of the fields 



* Treatise, Art. 689. 



t "The Induction of Currents in Cores." Electrician, 1884. Papers, Vol. i., 

 p. 353. 



t "On the Self-induction and Eesistance of Straight Conductors." Phil. Mag. 

 21, p. 381, 1886. 



" Ueber die Fortleitung elektrischer Wellen durch Drahte." Wied. Ann. 37, 

 p. 395, 1889. Jones's trans, p. 160. 



